If you're a white southerner, there's no difference. Both are an existential threat and it's both politically convenient and pretty sincere on their part to conflate antislavery and abolitionist Americans.

White northerners feel otherwise. I go a bit into the distinctions they drew, and which the abolitionists sometimes in turn drew against them, in that post:

So the antislavery side asked little and got nothing out of the so-called Missouri Compromise. This led to about a decade of disillusionment that for some coalesces in the conviction that the existing order cannot be counted on to end slavery, now or in the future. The most famous exponent of that viewpoint is William Lloyd Garrison, who eventually decided the Constitution was a proslavery document and the Union fundamentally a slavery-preservation machine. (The recent trend among historians, myself included, is to agree with him there.) So the strategy becomes to press for immediate, unconditional abolition everywhere...but also largely by means of moral suasion and fiery rhetoric rather than by working within the system which had betrayed them.

Antislavery people want to enact various policies through the federal government which indirectly end slavery by, as you say, putting it on the ultimate path to extinction. The idea is that slavery will work itself out, with just a little help to constrain it in its bounds. Abolitionists, broadly, think that's all a load of crap. The political process is fixed in favor of slavery, the parties are full of people who compromise with enslavers, and a radical disavowal of and disassociation with enslavers and enslaver-enablers is essential. Occasionally they go so far as to preach disunion.

So the big items for an abolitionist, bearing in mind that this is kind of an ideal type and lots of people who consider themselves abolitionists aren't consistent, would run something like this:

1. Abolition is immediate. It happens right now, not over the course of decades or centuries.
2. Abolition is uncompensated, unlike previous plans that "bought" slaves from their enslavers by binding them to a term of service before freedom and/or offering up actual cash.
3. The Constitution, the national parties, and the United States themselves are all functionally and probably intentionally proslavery, so they cannot be trusted and mainstream politics must be abjured.
4. That means either the abolitionists must organize their own, pure organizations like the Liberty Party (a spoiler party that managed to help deliver James K. Polk to the presidency) or through more radical means yet, such as by ending the union with the slave states. This also means that abolitionist groups spend a regrettable amount of time feuding amongst themselves over who is doing things the most right.
5. Some prominent abolitionists, like Garrison, are outright pacifists. Others believe violence counterproductive. But since political means are largely not on the table, the preferred tactic for attacking slavery where it is is moral suasion. Some kind of moral revolution needs to take place inside the enslaver's heart, analogous to a religious conversion. (Many white antislavery and abolitionist Americans came to their own convictions by a process they considered similar.)
6. Abolitionist discourse is, as the previous suggests, much more religious, dwelling on the suffering of the slaves at the hands of the enslavers and the moral turpitude inherent in enslaving people as a menace to evangelical, bourgeoisie mores. Abolitionists are less concerned with whether or not slavery is holding the South or the nation back economically or is a threat to white republicanism.
7. Abolitionists tend to be reformers and radicals in general, if in a middle class, nineteenth century way. An abolitionist is more likely to be a temperance person and otherwise support the moral reform movements of the era, even if their primary focus is against slavery. Antislavery people do some of that as well, but overall tend more latitudinarian.
8. Abolitionists are much more likely to be in favor of some forms of social and political equality between blacks and whites, though this remains somewhat controversial within the movement. Antislavery Americans, at least until the Civil War, get tarred with the brush of egalitarianism and tend to react with hostility to the suggestion.

That said, I want to stress again that these are generalizations and commitment to them by abolitionists in good standing can be very situational. Non-violence may be fine for preaching against direct attacks on enslavers in their persons, but abolitionists might turn out to stop the rendition of an accused fugitive and threaten or give violence to do so. Few are going to be too upset if someone else does, though they'd probably say they preferred otherwise in most cases. Many were a-ok with leading United States Colored Troops formations too.